Question

Question #cdf3d

Answer 1

`16 x^2 - 24 x y + 9 y^2 - 144 x + 8 y + 224 = 0`

Explanation:

The formula for the Distance from a Point to a Line is:

`d = |ax+by+c|/sqrt(a^2+b^2)" [1]"`

where `(x,y)` is the point and the line is of the form `ax+by+c = 0`.

The distance from any point `(x,y)` on the parabola to the line `3x+4y-1= 0` is:

`d = |3x+4y-1|/sqrt(3^2+4^2)`

`d = |3x+4y-1|/5" [2]"`

The distance from the focus `(3,0)` to any point, `(x,y)` on the parabola is:

`d = sqrt((x-3)^2+(y-0)^2)" [3]"`

The definition of a parabola is the locus of points equidistant from its directrix and its focus, therefore, we can derive the equation of the parabola by setting the right side of equation [2] equal to the right side of equation [3]:

`|3x+4y-1|/5 = sqrt((x-3)^2+(y-0)^2)`

Multiply both sides by 5:

`|3x+4y-1| = 5sqrt((x-3)^2+(y-0)^2)`

Square both sides:

`(3x+4y-1)^2 = 25((x-3)^2+(y-0)^2)`

Expand the squares:

`9x^2 + 24xy + 16y^2 - 6x - 8y + 1 = 25x^2 + 25y^2 - 150x + 225`

Simplify:

`-16 x^2 + 24 x y - 9 y^2 + 144 x - 8 y - 224 = 0`

Multiply both sides by -1:

`16 x^2 - 24 x y + 9 y^2 - 144 x + 8 y + 224 = 0`

The above is the standard Cartesian form for a conic section.

Here is a graph of the parabola, the focus and the directrix: